I used to feel that you had to have calculus to truly learn physics, but I've grown to think the opposite. There's a surprising amount of physics that you can learn without calculus. The level of math required depends on the level of the course. Do high school freshmen really need to know how to calculate the moment of inertia of various solids, or is it more important that they understand what rotational mass means? I'd rather they understand why the solid disk rolls down the slope faster than the hollow ring than being able to calculate the moment of inertia of either shape. On the other hand, for the physics or engineering majors in college, learning how to set up the integral is a technique they definitely need to learn because they'll be doing similar calculations later on. (Plus they can use all the practice they can get in evaluating the integral as well.)
Also, you have to take into account the goal of a course. The life science majors who take physics aren't planning to become physicists. Is it really important for them to get a true feeling for what it really means to do physics, or is the goal to expose them to the ideas of physics so they have a decent understanding of what we currently know and what we don't? I'd say the latter is more important than the former for this group of students.
I do understand and sympathize with those who object to removing math from a course simply because the students are afraid of math–that is, dumbing down the course. If a course has a calculus prerequisite, there's no valid reason to avoid using it in class when it makes the exposition clearer, or expecting students to solve a few problems that require calculus.
When you major in physics in the US, you typically don't learn, say, classical mechanics once. That's not how people learn. Most of us learn through repeated exposure to ideas and concepts. You see classical mechanics in high school physics, in intro physics in college, in an upper division course, and again in grad school. Each time the level of sophistication grows. You wouldn't start with teaching Lagrangians and Hamiltonians to high school students because it would be pointless. They simply wouldn't get it.
So getting back to the original point of this thread, the same considerations apply to learning calculus. Does the student want to learn the basic ideas of calculus, how to apply them, and how to do the calculations? Or should he or she immerse themselves into proving every last detail to get a true feeling for what real mathematicians do? For someone starting out, I'd lean toward the former. Ideally students should understand the reasoning used to reach various results, but I don't think it's particularly useful to spend a lot of time at this level trying to write proofs for everything.
. When you say calculus, you probably mean what's in Spivak/Apostol or some analysis books which I did not tackle until later. When I say calculus, I kind of mean the calculus necessary to do physics.